Finite Group Databasebeta

← C2×#24850 #6445618

Subgroups — C2×#24850 #6445618

Order 512.

2276 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#479C211yesmaximal, characteristic, complemented
2C22×#424C211yesmaximal, characteristic
3C2×#2629C211yesmaximal, characteristic
4C2×#2323C211yesmaximal, characteristic
5#24850C2116yesmaximal, direct product factor
6C2×#2088C211yesmaximal, characteristic, complemented
7C2×#2086C211yesmaximal, characteristic, complemented
8C2×#2068C211yesmaximal, characteristic, complemented
9C2×#2068C211yesmaximal, characteristic, complemented
10C2×#1989C211yesmaximal, characteristic, complemented
11C2×#1947C211yesmaximal, characteristic, complemented
12C2×#1894C211yesmaximal, characteristic
13C2×#1892C211yesmaximal, characteristic
14C2×#1809C211yesmaximal, characteristic
15C2×#1808C211yesmaximal, characteristic
16C2×#1803C211yesmaximal, characteristic, complemented
17C23×M16C2211yescharacteristic
18C24×C8C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C22×C22⋊Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C2×#479—28no—
24C2×#479C2218yescomplemented
25C22×#107C2211yescharacteristic
26C22×#107C2211yescharacteristic
27C22×#107C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C2×#424—28no—
30C2×#424C2218yes—
31C22×C4⋊C8C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C22×C4⋊C8C2211yescharacteristic
34C22×C4⋊C8C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#103C2211yescharacteristic
37C22×#103C2211yescharacteristic
38C22×#103C2211yescharacteristic
39#2629—24no—
40#2629C2218yes—
41C2×#398—21no—
42C2×#398—21no—
43C2×#396—21no—
44C2×#396—21no—
45#2323—24no—
46#2323C2218yes—
47C2×#391C2211yescharacteristic
48C2×#389—21no—
49C2×#389C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#388—21no—
53C2×#388—21no—
54C2×#387—21no—
55C2×#384C2211yescharacteristic
56C2×#380C2211yescharacteristic
57C2×#380C2211yescharacteristic
58C2×#380C2211yescharacteristic
59C2×C4×C4⋊C4C2211yescharacteristic
60#2088C2218yescomplemented
61#2086C2218yescomplemented
62#2068C2218yescomplemented
63#2068C2218yescomplemented
64#1989C2218yescomplemented
65#1947C2218yescomplemented
66#1894C2218yes—
67#1892C2218yes—
68#1809C2218yes—
69#1808C2218yes—
70#1803C2218yescomplemented
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#340C2211yescharacteristic
78C2×#340C2211yescharacteristic
79C23×Q8C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C22×M16—24no—
82C22×M16—24no—
83C22×M16—42no—
84C22×M16C2314yes—
85C23×C8—24no—
86C23×C8—24no—
87C23×C8—42no—
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90C23×C8C2314yes—
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C2×#125—22no—
94C2×#125—24no—
95C2×#125—24no—
96C2×#125C2314yes—
97C2×C22⋊Q8—22no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8Q818yes—
100C2×C22⋊Q8C2314yes—
101C2×C4×Q8—22no—
102C2×C4×Q8—24no—
103C2×C4×Q8—24no—
104C2×C4×Q8C2314yes—
105C2×#117—24no—
106C2×#117—24no—
107C2×#117D414yes—
108C2×#117C2314yes—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4D411yescharacteristic
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4D411yescharacteristic
132C22×C22⋊C4D411yescharacteristic
133C22×C42—21no—
134C22×C42—21no—
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#479—232no—
138C2×#107—24no—
139C2×#107—42no—
140C2×#107—42no—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#107C2314yes—
144C2×#106—24no—
145C2×#106C2314yes—
146#424—232no—
147C2×C4⋊C8—24no—
148C2×C4⋊C8—24no—
149C2×C4⋊C8—24no—
150C2×C4⋊C8D418yes—
151C2×C4⋊C8C2314yes—
152C2×C4⋊C8C2314yes—
153C2×C4⋊C8C2314yes—
154C2×C4⋊C8C2314yes—
155C2×#103—24no—
156C2×#103—24no—
157C2×#103—42no—
158C2×#103—42no—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163#398—28no—
164#398—28no—
165#396—28no—
166#396—28no—
167#391—22no—
168#391C2314yes—
169#389—28no—
170#389—22no—
171#389—22no—
172#389—22no—
173#389C2314yes—
174#389C2314yes—
175#389C2314yes—
176#388—28no—
177#388—28no—
178#387—28no—
179#384—22no—
180#384C2314yes—
181#380—22no—
182#380—22no—
183#380—22no—
184#380C2314yes—
185#380C2314yes—
186#380C2314yes—
187C4×C4⋊C4—22no—
188C4×C4⋊C4C2314yes—
189C2×#95—21no—
190C2×#95—21no—
191C2×#95—21no—
192C2×#95—21no—
193C2×#95—21no—
194C2×#95—21no—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206C25C2×D411yescharacteristic
207C22×Q8—21no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8C2×Q814yes—
214C22×Q8C2412yes—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D412yes—
232C23×C4C2×D411yescharacteristic
233C23×C4C2×Q814yes—
234C23×C4#4114yes—
235C23×C4C2412yes—
236C23×C4C2411yescharacteristic
237C2×M16—28no—
238C2×M16—28no—
239C2×M16—44no—
240C2×M16—416no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—28no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8—22no—
252C22×C8—28no—
253C22×C8—44no—
254C22×C8—416no—
255C22×C8C2×D414yes—
256C22×C8C2×D414yes—
257C22×C8C2412yes—
258C22×C8C2412yes—
259C22×C8C2412yes—
260C22×C8C2412yes—
261#125—28no—
262#125—28no—
263#125—48no—
264C22⋊Q8—216no—
265C22⋊Q8—28no—
266C22⋊Q8C2×Q8116yes—
267C4×Q8—28no—
268C4×Q8—28no—
269C4×Q8—48no—
270#117—216no—
271#117—28no—
272#117—28no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—82no—
309C2×C4⋊C4—82no—
310C2×C4⋊C4—82no—
311C2×C4⋊C4—82no—
312C2×C4⋊C4—82no—
313C2×C4⋊C4C2×D414yes—
314C2×C4⋊C4C2×D414yes—
315C2×C4⋊C4C2×Q814yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C22⋊C4—28no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—82no—
331C2×C22⋊C4—82no—
332C2×C22⋊C4—82no—
333C2×C22⋊C4C2×D414yes—
334C2×C22⋊C4C2×D414yes—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—44no—
340C2×C42—82no—
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2×D414yes—
344C2×C42#4114yes—
345C2×C42C2412yes—
346C2×C42C2412yes—
347#107—28no—
348#107—44no—
349#107—44no—
350#106—28no—
351C4⋊C8—28no—
352C4⋊C8—28no—
353C4⋊C8—28no—
354C4⋊C8C2×D4116yes—
355#103—28no—
356#103—28no—
357#103—44no—
358#103—44no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24C22⋊Q814yes—
375C24C22×D412yes—
376C24C22×D411yescharacteristic
377C2×Q8—24no—
378C2×Q8—22no—
379C2×Q8—28no—
380C2×Q8—28no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8C22×Q814yes—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4#12114yes—
445C22×C4#12114yes—
446C22×C4C22⋊Q814yes—
447C22×C4C22⋊Q818yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×Q814yes—
456C22×C4C22×Q812yes—
457C22×C4C2×#4114yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4H(5,𝔽2)12yes—
461C22×C42−1+412yes—
462C22×C4C2511yescharacteristic
463M16—416no—
464C2×C8—28no—
465C2×C8—28no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—416no—
477C2×C8—44no—
478C2×C8—44no—
479C2×C8C22×D414yes—
480C2×C8C22×D414yes—
481C4⋊C4—28no—
482C4⋊C4—28no—
483C4⋊C4—216no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—48no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—84no—
508C4⋊C4—84no—
509C4⋊C4—84no—
510C4⋊C4—84no—
511C4⋊C4—84no—
512C4⋊C4C22×Q814yes—
513C22⋊C4—216no—
514C22⋊C4—28no—
515C22⋊C4—28no—
516C22⋊C4—84no—
517C22⋊C4—84no—
518C22⋊C4—84no—
519C42—24no—
520C42—22no—
521C42—22no—
522C42—48no—
523C42—84no—
524C42#12118yes—
525C42C22×D414yes—
526C42C2×#4114yes—
527C23—28no—
528C23—28no—
529C23—22no—
530C23—28no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23C2×#12112yes—
541C23C2×C22⋊Q814yes—
542C23C2×C22⋊Q812yes—
543C23#53712yes—
544C23#57612yes—
545C23#57712yes—
546C23#57812yes—
547C23#57812yes—
548C23C23×D411yescharacteristic
549Q8—24no—
550Q8—24no—
551Q8—48no—
552Q8—42no—
553Q8—42no—
554C2×C4—22no—
555C2×C4—28no—
556C2×C4—22no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—216no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—48no—
577C2×C4—42no—
578C2×C4—48no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—84no—
603C2×C4—84no—
604C2×C4—84no—
605C2×C4—84no—
606C2×C4—84no—
607C2×C4—84no—
608C2×C4C2×#12114yes—
609C2×C4C2×#12114yes—
610C2×C4C2×#12112yes—
611C2×C4C2×C22⋊Q814yes—
612C2×C4C2×C22⋊Q812yes—
613C2×C4#53712yes—
614C2×C4#54614yes—
615C2×C4D4×Q814yes—
616C2×C4#55214yes—
617C2×C4#55414yes—
618C2×C4C23×D411yescharacteristic
619C2×C4C23×Q811yescharacteristic
620C2×C4C22×#4111yescharacteristic
621C2×C4C22×#4111yescharacteristic
622C2×C4C2×H(5,𝔽2)11yescharacteristic
623C2×C4C2×2−1+411yescharacteristic
624C8—24no—
625C8—24no—
626C8—42no—
627C8—42no—
628C22—24no—
629C22—24no—
630C22—24no—
631C22—416no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22#241714yes—
637C22#306114yes—
638C22#306314yes—
639C22#308214yes—
640C22#308314yes—
641C22#325714yes—
642C22#326114yes—
643C22C22×#12111yescharacteristic
644C22C22×C22⋊Q811yescharacteristic
645C22C2×#53711yescharacteristic
646C22C2×#57611yescharacteristic
647C22C2×#57711yescharacteristic
648C22C2×#57811yescharacteristic
649C22C2×#57811yescharacteristic
650C4—24no—
651C4—24no—
652C4—24no—
653C4—42no—
654C4—42no—
655C4—44no—
656C4—44no—
657C4—82no—
658C4—82no—
659C4—82no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4#241714yes—
664C4C2×#54611yescharacteristic
665C4C2×D4×Q811yescharacteristic
666C4C2×#55211yescharacteristic
667C4C2×#55411yescharacteristic
668C2—44no—
669C2#2485018yesdirect product factor
670C2C2×#241711yescharacteristic
671C2C2×#306111yescharacteristic
672C2C2×#306311yescharacteristic
673C2C2×#308211yescharacteristic
674C2C2×#308311yescharacteristic
675C2C2×#325711yescharacteristic
676C2C2×#326111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C22×#479C2semidirect
5. #24850C2direct (N × H)
6. C2×#2088C2semidirect
7. C2×#2086C2semidirect
8. C2×#2068C2semidirect
9. C2×#2068C2semidirect
10. C2×#1989C2semidirect
11. C2×#1947C2semidirect
16. C2×#1803C2semidirect
24. C2×#479C22semidirect
60. #2088C22semidirect
61. #2086C22semidirect
62. #2068C22semidirect
63. #2068C22semidirect
64. #1989C22semidirect
65. #1947C22semidirect
70. #1803C22semidirect
669. C2#24850direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#4792
C22×#4242
C2×#26292
C2×#23232
#248502
C2×#20882
C2×#20862
C2×#20682
C2×#19892
C2×#19472
C2×#18942
C2×#18922
C2×#18092
C2×#18082
C2×#18032